2004 AMC 10B 第 14 题

Question

一个袋子最初只含红色和蓝色弹珠，且蓝色比红色多。向袋中加入红色弹珠，直到蓝色弹珠只占袋中弹珠的

\tfrac{1}{3}

。然后加入黄色弹珠，直到蓝色弹珠只占袋中弹珠的

\tfrac{1}{5}

。最后，将袋中蓝色弹珠的数量加倍。现在袋中弹珠有几分之几是蓝色？

A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only

\tfrac{1}{3}

of the marbles in the bag are blue. Then yellow marbles are added to the bag until only

\tfrac{1}{5}

of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue?

\dfrac{1}{5}

\dfrac{1}{4}

\dfrac{1}{3}

\dfrac{2}{5}

\dfrac{1}{2}

设蓝色弹珠数为

B

，追踪每一步的总数。

Let

B

be the number of blue marbles and track the total at each stage

在最后加倍之前，有

B

个蓝色弹珠，总数为

5B

。

Just before doubling there are

B

blue out of

5B

total

设蓝色弹珠数为

B

。加入红色弹珠后，总数为

3B

；加入黄色弹珠后，总数为

5B

，蓝色仍为

B

。

将蓝色弹珠数量加倍后，有

2B

个蓝色弹珠，总数变为

6B

，所以蓝色所占比例为

\dfrac{2B}{6B} = \dfrac{1}{3}

。

所以正确答案是 C。

Let there be

B

blue marbles. After adding red marbles the total is

3B;

after adding yellow marbles the total is

5B,

still with

B

blue.

Doubling the blue marbles gives

2B

blue out of

6B

total, which is

\dfrac{2B}{6B} = \dfrac{1}{3}.

Thus, the correct answer is C.

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